Quantum Electrodynamics Prof. John Schliemann, Sebastian Waeber SS 19

Sheet 12

Task 30: The regularized polarization function

On the last sheet we used Pauli-Villars regularization to write the regularized polari- reg 2 zation tensor iΠµν as the gauge invariant combination (q gµν − qµqν) times e2 Λ2 i ln + Π(¯ q 2), (1) 12π2 m2 where m is the mass of the electron, Λ is the momentum scale and Π¯ is a finite integral over the parameter β, introduced on the last sheet. More specifically Π¯ there turned out to be proportional to Z 1 dββ(1 − β) ln[1 − β(1 − β)q 2/m2]. (2) 0 In this task we are going to solve this integral. a) Perform a partial integration to get rid of the logarithm and introduce the new integration variable v = 2β − 1 to write the integrand as a rational function of v 2. b) Consider the integral Z 1 1 dv 2 . (3) 0 v − c In the regions c ≤ 0 and in the region 1 ≤ c you can solve this integral easily. For 0 < c < 1 we remember a trick, when treating Feynman-propagators: In order to treat poles on the integration curve we introduce a tiny imaginary part c → c + i, which we send to 0 in the end. We can split the integration curve into a term comming from the residuum theorem and a principle value integral. Derive the solution to (3) for the different regions of c. c) To treat the integral deduced in a), we consider Z 1 v n In = dv 2 . (4) 0 v − c − i

Find a recursiv relation between In and In−2 and use this to write down an explicit expression for Π¯ for the three cases considered in b) now with respect to q 2.

Task 31: The Uehling-Potential

reg The higher order correction to the polarization tensor Πµν gives rise to a quantum field theoretic correction to the classical Coulomb-potential, which is known as the Uehling potential. For this we consider the source

jµ = −Zeδ(~x)δµ0 (5)

1 of a stationary pointlike charge Z · e. For the first order (in α) corrected photon 1 propagator DF,µν(q):

1 reg DF,µν(q) = DF,µν(q) + DF,µa (q)Πab DF,bν(q) (6) we have the relation Z d 4q A0 (x) = e−iqx D 1 (q)j ν(q), (7) µ (2π)4 F,µν

ν with the elecromagnetic current j (q) and DF,µν(q) is the free photon propagator. Derive the Uehling potential using your results from the last sheet and the previous task. Give your result depending on the renormalized charge.

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